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Apr
26
comment Euler characteristic - reference question
Sullivan, in "Combinatorial invariants of analytic spaces" (1971), comes close to stating this result. To get the result from what Sullivan does, one needs to use that algebraic varieties have a Whitney stratification, and that a tubular neighborhood of a stratum is a fiber bundle whose fiber is a cone over the link. Sullivan's argument shows that the tubular neighborhood has vanishing Euler characteristic, and then the result follows from Mayer-Vietoris. So it does indeed seem plausible that the result over $\mathbb C$ was known to experts before Laumon's result in the $\ell$-adic case.
Apr
26
comment Euler characteristic - reference question
@Donu $\mathcal F=0$ has vanishing Euler characteristic with or without compact support.
Apr
23
comment Using Lefschetz duality in algebraic geometry
I can't think of a reference off hand. There is a pretty useful appendix to Peters-Steenbrink's book "Mixed Hodge structures" which treats Borel-Moore homology and compact support cohomology, maybe you can look if it's stated there with a reference. There's also a "six functors"-way to prove it but I don't know if you'd be happy with that.
Apr
23
answered Using Lefschetz duality in algebraic geometry
Apr
15
comment Two transfers for ramified or branched covers
The answer to your first question is yes. You might find it easier to find a reference for (or prove) the stronger statement that the cycle class map $\mathrm{CH}^k(X) \to H^{2k}(X)$ commutes with pushforwards for all $k$. I'm assuming the varieties are smooth. But this might not be the kind of answer you want since it basically avoids talking about line bundles, thinking instead about algebraic cycles; in particular, I don't know how to define a norm map for topological line bundles.
Apr
10
answered Philosophy behind cohomological representations
Mar
21
comment Does the “holomorphic spheres-to-continuous spheres” forgetful function respect the mixed Hodge structures on homotopy groups?
Jason, you're describing how Morgan gives a structure of mixed Hodge complex on the cdga minimal model of $X$, right? A comment is that to go from that to a MHS on homotopy groups you need moreover to know that a structure of mixed Hodge complex on a cdga gives a canonical such structure also on its bar construction, so $H^\bullet(\Omega X)$ gets a natural MHS. Taking primitives gives the MHS on homotopy groups. So regarding @VivekShende's question, it seems possible that the bar construction can be realized as the cohomology of a simplicial scheme built out of $X$.
Mar
19
comment Does the “holomorphic spheres-to-continuous spheres” forgetful function respect the mixed Hodge structures on homotopy groups?
I'd expect there to be a Tate twist at least. If $k=0$ you get a map from $H_0(H_\beta)=\mathbf Q(0)$ to $H_2(X)$, which has weight $-2$.
Mar
16
comment Eisenstein Series on Siegel Space
Since you said "anything would be welcome" I can mention Schwermer's "On Euler products and residual Eisenstein cohomology classes for Siegel modular varieties". At one point I had to think a bit about Eisenstein cohomology of $A_2$ and this paper was quite useful.
Mar
9
awarded  Enlightened
Mar
9
awarded  Nice Answer
Mar
8
comment A spectral sequence for computing cohomology of a space from that of its strata
Yes, it works in intersection cohomology as well, sort of. You can use the sheafy approach but plug in $\mathrm{IC}_X$ instead of $\mathbf Z_X$. The only issue is that if $i$ is the inclusion of a closed subspace $A$ then $i^\ast \mathrm{IC}_X$ is not in general equal to the intersection complex on $A$ (even with a degree shift), unlike $i^\ast \mathbf Z_X = \mathbf Z_A$. So you need to compute the sheaves $i^\ast \mathrm{IC}_X$ in order to get a spectral sequence. One example of such a calculation is Section 3.4 of Beilinson-Ginzburg-Soergel "Koszul duality patterns in representation theory".
Mar
8
revised A spectral sequence for computing cohomology of a space from that of its strata
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Mar
8
answered A spectral sequence for computing cohomology of a space from that of its strata
Mar
4
awarded  Popular Question
Mar
1
comment factorization of the cohomology of configuration space
Oh! That's very nice.
Feb
15
comment About the characteristic polynomial of Frobenius of the Jacobian of a genus 2 hyperelliptic curve
Anyway, expressing the coefficients of the characteristic polynomial in terms of the Frobenius eigenvalues has not so much to do with étale cohomology - it's an exercise about symmetric functions.
Feb
15
comment About the characteristic polynomial of Frobenius of the Jacobian of a genus 2 hyperelliptic curve
Hi, I don't know a great reference off the top of my head that doesn't go into much more advanced things than what you are probably interested in. Maybe if you google "Weil conjectures for curves" you'll find some worked out examples to get you started.
Feb
13
revised About the characteristic polynomial of Frobenius of the Jacobian of a genus 2 hyperelliptic curve
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Feb
12
answered About the characteristic polynomial of Frobenius of the Jacobian of a genus 2 hyperelliptic curve